In the deterministic map filter, we adopt a complementary
strategy and seek a deterministic transformation from the
forecast to
the
filtering distribution. We view the transformation as a composition
of two functions. The first function is a KR rearrangement that
pushes forward the forecast \(\pi _{X}\)  to a reference distribution.
This rearrangement can be estimated via convex optimization given only
forecast samples, and yields an implicit approximation of the forecast
density, \(\widehat{\pi }_{X}\) . The second function is a KR
rearrangement that pushes forward the reference to an approximation of
the filtering distribution given by \(\widehat{\pi }_{X\vert Y}\propto \pi _{Y\vert X} \, \widehat{\pi }_{X}\) , which can be
evaluated in closed form up to a normalizing constant. The estimation
of the latter rearrangement leads to a variational problem, first
proposed by [1]}, that is in general non-convex.
This is an important difference with respect to the stochastic map
filter, which relies entirely on convex optimization, at the cost of
estimating higher-dimensional rearrangements. Yet, in the special
case of local observations that are conditionally independent, we show
how to implement the deterministic map filter using only convex
optimization (Appendix ). The resulting scheme
has a flavor similar to the multivariate rank histogram filter [2]}. For a
linear parameterization of the estimators and Gaussian
likelihoods, the deterministic map filter reduces to a (deterministic)
square-root EnKF [3]}; otherwise, it is far more
general.
